## Group Actions, p Groups and Sylow Subgroups

### p Groups and Sylow Subgroups

For p prime, G is a p group if the order of every element in G is a power of p.
Note that infinite groups can be p groups.
The infinite direct product of **Z**p is an example.
If |G| is a power of p then every subgroup, including the powers of x,
has order dividing |G|, hence G is a p group.
Conversely, if some other prime q divides |G|, then G contains a q cycle, and is not a p group.
For finite groups, p group means |G| is a power of p.

A sylow subgroup is a maximal subgroup that is also a p group.
Use zorn's lemma to show such a group exists when the containing group is infinite.

Let the p group G have a normal subgroup H.
Since x*H/x is H, G acts on the set H by conjugation.
This action leaves the center of G fixed.
When restricted to H,
it fixes H intersect the center of G.
By the fixed point principle, this fixed set has size n,
where n = |H| mod p.
Of course H is a p group, so n = 0.
The center of G, intersected with H, is always divisible by p.

As a special case,
set H = G, and p divides the number of elements in the center of G.
Therefore the center of every p group is nontrivial.